# AP Calculus AB : Derivative defined as the limit of the difference quotient

## Example Questions

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### Example Question #1 : Derivative Defined As The Limit Of The Difference Quotient

What is the derivative of ?

Explanation:

To solve this problem, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.

Remember that anything to the zero power is one.

### Example Question #2 : Derivative Defined As The Limit Of The Difference Quotient

What is the derivative of ?

Explanation:

To solve this problem, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.

We're going to treat  as , as anything to the zero power is one.

That means this problem will look like this:

Notice that , as anything times zero is zero.

Remember, anything to the zero power is one.

### Example Question #2 : Finding Derivative Of A Function

What is the derivative of ?

Explanation:

To get , we can use the power rule.

Since the exponent of the  is , as , we lower the exponent by one and then multiply the coefficient by that original exponent:

Anything to the  power is .

### Example Question #3 : Derivative Defined As The Limit Of The Difference Quotient

Explanation:

To solve this equation, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.

We're going to treat  as  since anything to the zero power is one.

Notice that  since anything times zero is zero.

### Example Question #4 : Derivative Defined As The Limit Of The Difference Quotient

What is the derivative of ?

Explanation:

To solve this equation, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.

We're going to treat  as  since anything to the zero power is one.

Notice that  since anything times zero is zero.

That leaves us with .

Simplify.

As stated earlier, anything to the zero power is one, leaving us with:

### Example Question #32 : Calculus I — Derivatives

What is the derivative of ?

Explanation:

To solve this equation, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.

We're going to treat  as  since anything to the zero power is one.

Notice that  since anything times zero is zero.

Just like it was mentioned earlier, anything to the zero power is one.

### Example Question #5 : Derivative Defined As The Limit Of The Difference Quotient

What is the derivative of ?

Explanation:

To take the derivative of this equation, we can use the power rule. The power rule says that we lower the exponent of each variable by one and multiply that number by the original exponent.

Simplify.

Remember that anything to the zero power is equal to one.

### Example Question #25 : Derivatives

What is the derivative of ?

Explanation:

To take the derivative of this equation, we can use the power rule. The power rule says that we lower the exponent of each variable by one and multiply that number by the original exponent.

We are going to treat  as  since anything to the zero power is one.

Notice that  since anything times zero is zero.

Simplify.

As stated before, anything to the zero power is one.

### Example Question #26 : Derivatives

What is the derivative of ?

Explanation:

To find the first derivative, we can use the power rule. We lower the exponent on all the variables by one and multiply by the original variable.

Anything to the zero power is one.

### Example Question #6 : Derivative Defined As The Limit Of The Difference Quotient

What is the derivative of ?

Explanation:

To find the first derivative, we can use the power rule. We lower the exponent on all the variables by one and multiply by the original variable.

We're going to treat  as  since anything to the zero power is one.

For this problem that would look like this:

Notice that  since anything times zero is zero.

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